Reasoning system for reasoning with uncertainty

• US 4,860,213 A
• Filed: 10/01/1987
• Issued: 08/22/1989
• Est. Priority Date: 10/01/1987
• Status: Expired due to Term

• Current Assignee: General Electric Company
• Original Assignee: General Electric Company
• Inventors: Bonissone, Piero P.
• Primary Examiner(s): Pellinen, A. D.
• Assistant Examiner(s): WILLIAMS, HOWARD L
• Application Number:
  US07/103,465
• Time in Patent Office: 691 Days
• Field of Search: 364/513, 364/300
• US Class Current: 706/52
• CPC Class Codes: G06N 5/04 Inference or reasoning models
• CPC Class Codes: Y10S 706/90 Fuzzy logic

Abstract

The invention is directed to an automated rule-based reasoning with uncertainty system and method for practicing same. The system has a three layer structure composed of representation, inference and control layers. Rule firings and conclusions are computed in the inference layer in a manner to propagate uncertainty information through the system in accordance with predetermined formulas designated by the control layer. Rule structures and conclusions and uncertainty information associated therewith are represented to the user in the representation layer.

Claims

1. A method, for practice on a computer, for reasoning with uncertainty in a rule-based reasoning system comprising a plurality of rules each having a premise and a conclusion to be determined true upon finding the premise to be true, said premise comprising at least one clause including a variable to which a value can be assigned, said method comprising the steps of:

associating a value certainty interval with each value assigned to one of said rule premise variables, said value certainty interval having a lower and an upper bound respectively representative of amounts of confirmation and failure of refutation of the assigned value;

associating a sufficiency factor and a necessity factor with each said rule, said sufficiency factor representing a strength of belief that the rule conclusion is true given the rule premise is true, said necessity factor representing a strength of belief that the rule conclusion is false given the rule premise is false;

computing a premise certainty interval from the value certainty intervals respectively associated with the values assigned to the premise variables, a lower bound of said premise certainty interval being a first predetermined function of the respective value certainty interval lower bounds, an upper bound of said premise certainty interval being said first function of the respective value certainty interval upper bounds; and

firing one of said rules by computing a conclusion detachment certainty interval having a lower and an upper bound, said conclusion detachment interval lower bound being a second predetermined function of both said rule sufficiency factor and said premise certainty interval lower bound, the conclusion detachment upper bound being a third predetermined function of both said rule necessity factor and said premise certainty interval upper bound, said conclusion detachment interval lower and upper bounds respectively representing the amounts of confirmation and failure of refutation of the rule conclusion.

5. The reasoning with uncertainty method of claim 1 wherein said value uncertainty, premise certainty and conclusion detachment intervals are defined on an interval [0,1.0];

said premise certainty interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[b,B]=[T.sub.i (a.sub.1,a.sub.2, . . . ,a.sub.m),T.sub.i (A.sub.1,A.sub.2, . . . ,A.sub.m)]

where;

[b,B] is said premise certainty interval;

[a₁,A₁ ], [a₂,A₂ ], . . . ,[a_m,A_m ] are m said value certainty intervals associated with the values assigned to the premise variables; and

T_i is a triangular norm (T-norm) function selected from a first function set comprising the predetermined T-norm functions T₁, T₂ and T₃ ; and

said conclusion detachment interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[c,C]=[T.sub.i (s,b),N(T.sub.i (n,N(B)))]

where;

[c,C] is said conclusion detachment interval;

T_i is the T-norm function selected from said first function set;

s is said rule sufficiency factor;

n is said rule necessity factor;

[b,B] is said premise certainty interval; and

N is a negation operator such that N(α

)=1-α

.

6. The reasoning with uncertainty method of claim 5 wherein each said T-norm function T_i is selected from said first function set in accordance with an attitude toward uncertainty evaluation;

the T-norm function T₁ corresponding to a conservative attitude toward uncertainty evaluation;

the T-norm function T₂ corresponding to an intermediate attitude toward uncertainty evaluation; and

the T-norm function T₃ corresponding to a nonconservative attitude toward uncertainty evaluation.

9. The reasoning with uncertainty method of claim 1 wherein selected ones of said rules are each associated with a respective source of rule firing context information;

and

said method including the additional step of evaluating, prior to the firing of each said selected rule, said context information to determine whether an appropriate context exists for the rule firing.

2. A method, for practice on a computer, for reasoning with uncertainty in a rule-based reasoning system comprising a plurality of rules each having a premise and a conclusion to be determined true upon finding the premise to be true, said premise comprising at least one clause including a variable to which a value can be assigned, said method comprising the steps of:

associating a value certainty interval with each value assigned to one of said rule premise variables, said value certainty interval having a lower and an upper bound respectively representative of amounts of confirmation and failure of refutation of the assigned value;

associating a sufficiency factor and a necessity factor with each said rule, said sufficiency factor representing a strength of belief that the rule conclusion is true given the rule premise is true, said necessity factor representing a strength of belief that the rule conclusion is false given the rule premise is false;

computing a premise certainty interval from the value certainty intervals respectively associated with the values assigned to the premise variables, a lower bound of said premise certainty interval being a first predetermined function of the respective value certainty interval lower bounds, an upper bound of said premise certainty interval being said first function of the respective value certainty interval upper bounds; and

firing one of said rules by computing a conclusion detachment certainty interval having a lower and an upper bound, said conclusion detachment interval lower bound being a second predetermined function of both said rule sufficiency factor and said premise certainty interval lower bound, the conclusion detachment upper bound being a third predetermined function of both said rule necessity factor and said premise certainty interval upper bound, said conclusion detachment interval lower and upper bounds respectively representing the amounts of confirmation and failure of refutation of the rule conclusion, wherein upon the same conclusion being provided by more than one rule in a predetermined group of said rules, said method comprises the additional step of;

aggregating the same conclusions by computing a conclusion aggregation certainty interval having a lower bound that is a fourth predetermined function of the conclusion detachment interval lower bounds respectively computed for the rules reaching the same conclusion, an upper bound of said conclusion aggregation interval being said fourth predetermined function of the conclusion detachment interval upper bounds respectively computed for the rules reaching the same conclusion, said conclusion aggregation interval lower and upper bounds respectively representing the amounts of confirmation and failure of refutation of the aggregated conclusion.

3. The reasoning with uncertainty method of claim 2 wherein upon computing a plurality of said conclusion aggregation intervals for the same conclusion, said method comprises the additional step of:

computing a source consensus certainty interval by taking the intersection of said plurality of conclusion aggregation intervals.

4. The reasoning with uncertainty method of claim 2 including the additional steps of:

excluding a selected conclusion from said aggregating step even though said selected conclusion is the same as the aggregated conclusion, said selected conclusion having an associated certainty interval; and

computing a source consensus certainty interval by taking the intersection of the associated certainty interval of said selected conclusion with each said conclusion aggregation interval computed for the same conclusion as said selected conclusion.

7. The reasoning with uncertainty method of claim 2 wherein said value certainty, premise certainty, conclusion detachment and conclusion aggregation intervals are defined on an interval [0,1.0];

said premise certainty interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[b,B]=[T.sub.i (a.sub.1,a.sub.2, . . . ,a.sub.m),T.sub.i (A.sub.1,A.sub.2, . . . ,A.sub.m)]

where;

[b,B] is said premise certainty interval;

[a₁,A₁ ], [a₂,A₂ ], . . . ,[a_m,A_m ] are m said value certainty intervals associated with the values assigned to the premise variables; and

T_i is a triangular norm (T-norm) function selected from a first function set comprising the predetermined T-norm functions T₁, T₂ and T₃ ;

said conclusion detachment interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[c,C]=[T.sub.i (s,b),N(T.sub.i (n,N(B)))]

where;

[c,C] is said conclusion detachment interval;

T_i is the T-norm function selected from said first function set;

s is said rule sufficiency factor;

n is said rule necessity factor;

[b,B] is said premise certainty interval; and

N is a negation operator such that N(α

)=1-α

; and

said conclusion aggregation interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[d,D]=[S.sub.i (c.sub.1,c.sub.2, . . . ,c.sub.m),S.sub.i (C.sub.1,C.sub.2, . . . ,C.sub.m)]

where;

[d,D] is said conclusion aggregation interval;

[c₁,C₁ ], [c₂,C₂ ], . . . ,[c_m,C_m ] are m said conclusion detachment intervals respectively associated with the same conclusion being aggregated; and

S_i is a triangular conorm (T-conorm) function selected from a second function set comprising the predetermined T-conorm functions S₂, S₂.5 and S₃.

8. The reasoning with uncertainty method of claim 7 wherein each said T-norm function T_i is selected from said first function set in accordance with an attitude toward uncertainty evaluation, such that:

the T-norm function T₁ corresponds to a conservative attitude toward uncertainty evaluation;

the T-norm function T₂ corresponds to an intermediate attitude toward uncertainty evaluation; and

the T-norm function T₃ corresponds to a nonconservative attitude toward uncertainty evaluation;

each said T-conorm function S_i being selected in accordance with whether the rules providing the same conclusion being aggregated are characterized as being correlated, such that;

said T-conorm function S₂ is selected for aggregating the same conclusions provided by uncorrelated rules;

said T-conorm function S₃ is selected for aggregating the same conclusions provided by correlated rules; and

said T-conorm function S₂.5 is selected for aggregating the same conclusions being provided by rules characterized as being intermediate between correlated and uncorrelated.

10. A rule-based reasoning system for reasoning with uncertainty for practice on a computer, comprising:

a knowledge base portion comprising a plurality of system rules each having a premise and a conclusion to be determined true upon finding the premise true, said premise comprising at least one clause including a variable to which a value can be assigned;

a representation portion for representing to a system user a value certainty interval associated with each value assigned to one of said rule premise variables, said value certainty interval having a lower and an upper bound respectively representative of amounts of confirmation and failure of refutation of the assigned value;

said representation portion additionally representing to the user a sufficiency factor and a necessity factor associated with each said rule, said sufficiency factor representing a strength of belief that said rule conclusion is true given said rule premise is true, said necessity factor representing a strength of belief that said rule conclusion is false given said rule premise is false;

a control portion for selecting computational functions in accordance with which the value certainty interval associated with each said assigned value is propagated through said rule based system;

an inference portion for computing a premise certainty interval from the value certainty intervals respectively associated with the values assigned to the premise clauses, a lower bound of said premise certainty interval being a first computational function of the respective value certainty interval lower bounds, said first function being selected in said control portion, an upper bound of said premise certainty interval being said first function of the respective value certainty interval upper bounds; and

said inference portion firing each said rule by computing a conclusion detachment interval having a lower and an upper bound, the conclusion detachment interval lower bound being a second computational function of both said rule sufficiency factor and said premise certainty interval lower bound, the conclusion detachment interval upper bound being a third computational function of both said rule necessity factor and said premise certainty interval upper bound, said second and third functions being selected by said control portion, said conclusion detachment interval lower and upper bounds being respectively representative of the amounts of confirmation and failure of refutation of the rule conclusion.

14. The reasoning with uncertainty system of claim 10 wherein said value certainty, premise certainty and conclusion detachment intervals are defined on an interval [0,1.0];

said inference portion computing said premise certainty interval in accordance with the equation;

space="preserve" listing-type="equation">[b,B]=[T.sub.i (a.sub.1,a.sub.2, . . . ,a.sub.m),T.sub.i (A.sub.1,A.sub.2, . . . ,A.sub.m)]

where;

[b,B] is said premise certainty interval;

[a₁,A₁ ], [a₂,A₂ ], . . . ,[a_m,A_m ] are m said value certainty intervals associated with the values assigned to the premise variables; and

T_i is a triangular norm (T-norm) function selected from a first function set comprising the predetermined T-norm functions T₁, T₂ and T₃ ; and

said inference portion computing said conclusion detachment interval in accordance with the equation;

space="preserve" listing-type="equation">[c,C]=[T.sub.i (s,b),N(T.sub.i (n,N(B)))]

where;

[c,C] is said conclusion detachment interval;

T_i is the T-norm function selected from said first function set;

s is said rule sufficiency factor;

n is said rule necessity factor;

[b,B] is said premise certainty interval; and

N is a negation operator such that N(α

)=1-α

.

15. The reasoning with uncertainty system of claim 14 wherein said control portion selects each said T-norm function T_i for use by said inference portion in computing said premise certainty and conclusion detachment intervals;

each said T-norm function T_i corresponding to a different attitude toward uncertainty evaluation such that;

the T-norm function T₁ corresponds to a conservative attitude toward uncertainty evaluation;

the T-norm function T₂ corresponds to an intermediate attitude toward uncertainty evaluation; and

the T-norm function T₃ corresponds to a nonconservative attitude toward uncertainty evaluation.

18. The reasoning with uncertainty system of claim 10 wherein an ignorance measure is maintained in said representation portion and associated with each said value certainty, premise certainty and conclusion detachment interval, said ignorance measure being equal to the arithmetic difference between the upper and lower bound of the interval with which it is associated.

19. The reasoning with uncertainty system of claim 10 wherein selected ones of said rules are each associated with a respective source of rule firing context information;

and

said control portion evaluating, prior to the firing of each said selected rule, said context information to determine whether an appropriate context exists for the rule firing.

11. A rule-based reasoning system for reasoning with uncertainty for practice on a computer, comprising:

a knowledge base portion comprising a plurality of system rules each having a premise and a conclusion to be determined true upon finding the premise true, said premise comprising at least one clause including a variable to which a value can be assigned;

a representation portion for representing to a system user a value certainty interval associated with each value assigned to one of said rule premise variables, said value certainty interval having a lower and an upper bound respectively representative of amounts of confirmation and failure of refutation of the assigned value;

said representation portion additionally representing to the user a sufficiency factor and a necessity factor associated with each said rule, said sufficiency factor representing a strength of belief that said rule conclusion is true given said rule premise is true, said necessity factor representing a strength of belief that said rule conclusion is false given said rule premise is false;

a control portion for selecting computational functions in accordance with the value certainty interval associated with each said assigned value is propagated through said rule based system;

an inference portion for computing a premise certainty interval from the value certainty intervals respectively associated with the values assigned to the premise clauses, a lower bound of said premise certainty interval being a first computational function of the respective value certainty interval lower bounds, said first function being selected in said control portion, an upper bound of said premise certainty being said first function of the respective value certainty interval upper bounds; and

said inference portion firing each said rule by computing a conclusion detachment interval having a lower and an upper bound, a conclusion detachment interval lower bound being a second computational function of both said rule sufficiency factor and said premise certainty interval lower bound, the conclusion detachment interval upper bound being a third computational function of both said rule necessity factor and said premise certainty interval upper bound, said second and third functions being selected by said control portion, said conclusion detachment interval lower and upper bounds being respectively representative of the amounts of confirmation and failure of refutation of the rule conclusion, and wherein said inference portion is adapted to compute a conclusion aggregation certainty interval for the same conclusion provided by more than one rule in a predetermined group of said rules, a lower bound of said conclusion aggregation interval being a fourth computational function of the conclusion detachment interval lower bounds respectively computed for the rules providing the same conclusion, an upper bound of said conclusion aggregation interval being said fourth function of the conclusion detachment interval upper bounds respectively computed for the rules providing the same conclusion, said fourth function being selected by said control portion, said conclusion aggregation interval lower and upper bounds being respectively representative of the amount of confirmation and failure of refutation of the aggregated conclusion.

12. The reasoning with uncertainty system of claim 11 wherein said inference portion is adapted to compute a source consensus certainty interval for a plurality of said conclusion aggregation intervals for the same conclusion, said source consensus interval being the intersection of said plurality of conclusion aggregation intervals.

13. The reasoning with uncertainty system of claim 11 wherein a selected conclusion can be excluded from the computation of said conclusion aggregation interval, said selected conclusion having an associated certainty interval;

and

said inference portion being adapted to compute a source consensus certainty interval that is the intersection of the associated certainty interval of said selected conclusion with each said conclusion aggregation interval computed for the same conclusion as said selected conclusion.

16. The reasoning with uncertainty method of claim 11 wherein said value certainty, premise certainty, conclusion detachment and conclusion aggregation intervals are defined on an interval [0,1.0];

said inference portion computing said premise certainty interval in accordance with the equation;

space="preserve" listing-type="equation">[b,B]=[T.sub.i (a.sub.1,a.sub.2, . . . ,a.sub.m),T.sub.i (A.sub.1,A.sub.2, . . . ,A.sub.m)]

where;

[b,B] is said premise certainty interval;

[b₁,A₁ ], [a₂, A₂ ], . . . , [a_m,A_m ] are m said value certainty intervals associated with the values assigned to the premise variables; and

T_i is a triangular norm (T-norm) function selected from a first function set comprising

the predetermined T-norm functions T₁, T₂ and T₃ ;

said inference portion computing said conclusion detachment interval in accordance with the equation;

space="preserve" listing-type="equation">[c,C]=[T.sub.i (s,b),N(T.sub.i (n,N(B))]

where;

[c,C] is said conclusion detachment interval;

T_i is the T-norm function selected from said first function set;

s is said rule sufficiency factor;

n is said rule necessity factor;

[b,B] is said premise certainty interval; and

N is a negation operator such that N(α

)=1-α

; and

said inference portion computing said conclusion aggregation interval in accordance with the equation;

space="preserve" listing-type="equation">[d,D]=[S.sub.i (c.sub.1,c.sub.2, . . . ,c.sub.m),S.sub.i (C.sub.1, C.sub.2, . . . ,C.sub.m)]

where;

[d,D] is said conclusion aggregation interval;

[c₁,C₁ ], [c₂,C₂ ], . . . , [C_m, C_m ] are m said conclusion detachment intervals respectively associated with the same conclusion being aggregated; and

S_i is a triangular conorm (T-conorm) function selected from a second function set comprising the predetermined T-conorm functions S₂, S₂.5 and S₃.

17. The reasoning with uncertainty system of claim 16 wherein said control portion selects each said T-norm function T_i for use by said inference portion in computing said premise certainty and conclusion detachment intervals, each said T-norm function T_i corresponding to a different attitude toward uncertainty evaluation such that:

the T-norm function T₁ corresponds to a conservative attitude toward uncertainty evaluation;

the T-norm function T₂ corresponds to an intermediate attitude toward uncertainty evaluation; and

the T-norm function T₃ corresponds to a nonconservative attitude toward uncertainty evaluation.

said control portion selecting each said T-conorm function S_i for use by said inference portion in computing said conclusion aggregation interval, each said T-conorm function S_i being selected in accordance with whether the rules providing the same conclusion being aggregated are characterized as being correlated, such that;

said T-conorm function S₂ is selected for aggregating the same conclusions provided by uncorrelated rules;

said T-conorm function S₃ is selected for aggregating the same conclusions provided by correlated rules; and

said T-conorm function S₂.5 is selected for aggregating the same conclusions being provided by rules characterized as being intermediate between correlated and uncorrelated.

20. A system, for practice on a computer, for reasoning with uncertainty in a rule-based reasoning system comprising a plurality of rules each having a premise and a conclusion to be determined true upon finding the premise to be true, said premise comprising at least one clause including a variable to which a value can be assigned, said reasoning system comprising:

means for associating a value certainty interval with each value assigned to one of said rule premise variables, said value certainty interval having a lower and an upper bound respectively representative of amounts of confirmation and failure of refutation of the assigned value;

means for associating a sufficiency factor and a necessity factor with each said rule, said sufficiency factor representing a strength of belief that the rule conclusion is true given the rule premise is true, said necessity factor representing a strength of belief that the rule conclusion is false given the rule premise is false;

means for computing a premise certainty interval from the value certainty intervals respectively associated with the values assigned to the premise variables, a lower bound of said premise certainty interval being a first predetermined function of the respective value certainty interval lower bounds, an upper bound of said premise certainty interval being said first function of the respective value certainty interval upper bounds; and

means for firing each of said rules by computing a conclusion detachment certainty interval having a lower and an upper bound, said conclusion detachment interval lower bound being a second predetermined function of both said rule sufficiency factor and said premise certainty interval lower bound, the conclusion detachment upper bound being a third predetermined function of both said rule necessity factor and said premise certainty interval upper bound, said conclusion detachment interval lower and upper bounds respectively representing the amounts of confirmation and failure of refutation of the rule conclusion.

24. The reasoning with uncertainty system of claim 20 wherein said value certainty, premise certainty and conclusion detachment intervals are defined on an interval [0,1.0];

said premise certainty interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[b,B]=[T.sub.i (a.sub.1,a.sub.2, . . . ,a.sub.m),T.sub.i (A.sub.1,A.sub.2, . . . ,A.sub.m)]

where;

[b,B] is said premise certainty interval;

[a₁,A₁ ], [a₂,A₂ ], . . . , [a_m, A_m ] are m said value certainty intervals associated with the values assigned to the premise variables; and

T_i is a triangular norm (T-norm) function selected from a first function set comprising the predetermined T-norm functions T₁, T₂ and T₃ ; and

said conclusion detachment interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[c,C]=[T.sub.i (s,b),N(T.sub.i (n,N(B)))]

where;

26. The reasoning with uncertainty system of claim 24 wherein each said T-norm function T_i is selected from said first function set in accordance with an attitude toward uncertainty evaluation;

the T-norm function T₁ corresponding to a conservative attitude toward uncertainty evaluation;

the T-norm function T₂ corresponding to an intermediate attitude toward uncertainty evaluation; and

the T-norm function T₃ corresponding to a nonconservative attitude toward uncertainty evaluation.

28. The reasoning with uncertainty system of claim 26 wherein each said T-norm function T_i is selected from said first function set in accordance with an attitude toward uncertainty evaluation, such that:

the T-norm function T₁ corresponds to a conservative attitude toward uncertainty evaluation;

the T-norm function T₂ corresponds to an intermediate attitude toward uncertainty evaluation; and

the T-norm function T₃ corresponds to a nonconservative attitude toward uncertainty evaluation;

each said T-conorm function S_i being selected in accordance with whether the rules providing the same conclusion being aggregated are characterized as being correlated, such that;

said T-conorm function S₂ is selected for aggregating the same conclusions provided by uncorrelated rules;

said T-conorm function S₃ is selected for aggregating the same conclusions provided by correlated rules; and

said T-conorm function S₂.5 is selected for aggregating the same conclusions being provided by rules characterized as being intermediate between correlated and uncorrelated.

29. The reasoning with uncertainty system of claim 20 wherein selected ones of said rules are each associated with a respective source of rule firing context information;

and

said system additionally including means for evaluating, prior to the firing of each said selected rule, said context information to determine whether an appropriate context exists for the rule firing.

21. A system, for practice on a computer, for reasoning with uncertainty system in a rule-based reasoning system comprising a plurality of rules each having a premise and a conclusion to be determined true upon finding a premise to be true, said premise comprising at least one clause including a variable to which a value can be assigned, said reasoning system comprising:

means for associating a value certainty interval with each value assigned to one of said rule premise variables, said value certainty interval having a lower and an upper bound respectively representative of amounts of confirmation and failure of refutation of the assigned value;

means for associating a sufficiency factor and a necessity factor with each said rule, said sufficiency factor representing a strength of belief that the rule conclusion is true given the rule premise is true, said necessity factor representing a strength of belief that the rule conclusion is false given the rule premise is false;

means for computing a premise certainty interval from the value certainty intervals respectively associated with the values assigned to the premise variables, a lower bound of said premise certainty interval being a first predetermined function of the respective value certainty interval lower bounds, an upper bound of said premise certainty interval being said first function of the respective value certainty interval upper bounds; and

means for firing each of said rules by computing a conclusion detachment certainty interval having a lower and upper bound, said conclusion detachment interval lower bound being a second predetermined function of both said rule sufficiency factor and said premise certainty interval lower bound, the conclusion detachment upper bound being a third predetermined function of both said rule necessity factor and said premise certainty interval upper bound, said conclusion detachment interval lower and upper bounds respectively representing the amounts of confirmation and failure of refutation of the rule conclusion, and wherein upon the same conclusion being provided by more than one rule in a predetermined group of said rules, said system additionally comprises;

means for aggregating the same conclusions by computing a conclusion aggregation certainty interval having a lower bound that is a fourth predetermined function of the conclusion detachment interval lower bounds respectively computed for the rules reaching the same conclusion, an upper bound of said conclusion aggregation interval being said fourth predetermined function of the conclusion detachment interval upper bounds respectively computed for the rules reaching the same conclusion, said conclusion aggregation interval lower and upper bounds being respectively representative of the amount of confirmation and failure of refutation of the aggregated conclusion.

22. The reasoning with uncertainty system of claim 21 wherein upon computing a plurality of said conclusion aggregation intervals for the same conclusion, said system further comprises:

means for computing a source consensus certainty interval by taking the intersection of said plurality of conclusion aggregation intervals.

23. The reasoning with uncertainty system of claim 21 further comprising:

means for excluding a selected conclusion from said aggregating step even though said selected conclusion is the same as the aggregated conclusions, said selected conclusion having an associated certainty interval; and

means for computing a source consensus certainty interval by taking the intersection of the associated certainty interval of said selected conclusion with each said conclusion aggregation interval computed for the same conclusion.

27. The reasoning with uncertainty system of claim 21 wherein said value certainty, premise certainty, conclusion detachment and conclusion aggregation intervals are defined on an interval [0,1.0];

said premise certainty interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[b,B]=[T.sub.i (a.sub.1,a.sub.2, . . . ,a.sub.m),T.sub.i (A.sub.1,A.sub.2, . . . ,A.sub.m)]

where;

[b,B] is said premise certainty interval;

[a₁,A₁ ], [a₂,A₂ ], . . . , [a_m,A_m ] are m said value certainty intervals associated with the values assigned to the premise variables; and

T_i is a triangular norm (T-norm) function selected from a first function set comprising the predetermined T-norm functions T₁, T₂ and T₃ ;

said conclusion detachment interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[c,C]=[T.sub.i (s,b),N(T.sub.i (n,N(B)))]

where;

[c,C] is said conclusion detachment interval;

T_i is the T-norm function selected from said first function set;

s is said rule sufficiency factor;

n is said rule necessity factor;

[b,B] is said premise certainty interval; and

N is a negation operator such that N(α

)=1-α

; and

said conclusion aggregation interval being computed in accordance with the equation;

space="preserve" listing-type="equation">[d,D]=[S.sub.i (c.sub.1,c.sub.2, . . . ,c.sub.m),S.sub.i (C.sub.1,C.sub.2, . . . ,C.sub.m)]

where;

[d,D] is said conclusion aggregation interval;

[c₁,C₁ ], [c₂,C₂ ], . . . , [c_m,C_m ] are m said conclusion detachment intervals respectively associated with the same conclusion being aggregated; and

S_i is a triangular conorm (T-conorm) function selected from a second function set comprising the predetermined T-conorm functions S₂, S₂.5 and S₃.

25. [c,C] is said conclusion detachment interval;

T_i is the T-norm function selected from said first function set;

s is said rule sufficiency factor;

n is said rule necessity factor;

[b,B] is said premise certainty interval; and

N is a negation operator such that N(α

)=1-α

.